Optimal Control Systems EE 622 Assignment 3 1. Describe Weistrass-Erdmann corner conditions for strong minima. 2. Minimize J(x, u) Z J(x, u) = 1 u2 dt (1) 0 subject to ẋ = −2x + u (2) with x(0) = 1 and x(1) = 0. Is your solution unique ? 3. A first order system is given by ẋ(t) = ax(t) + bu(t) (3) and the performance index is Z 1 tf (qx2 (t) + ru2 (t))dt (4) J= 2 0 where, x(t0 ) = x0 and x(tf ) is free and tf being fixed. Show that the optimal state x∗ (t) is given by sinh β(tf − t) x∗ (t) = x0 (5) sinh βtf p where, β = a2 + b2 q/r 4. Find the optimal control u∗ (t) for the plant which drives the system x˙1 (t) = x2 (t) (6) x˙2 (t) = −2x1 (t) + 5u(t) (7) with x1 (0) = 3, x2 (0) = 5, x1 (2) = 0 and x2 (0) = 0 which minimizes the the performance index given by Z 1 2 2 J= (x1 (t) + u2 (t))dt (8) 2 0 5. Find the optimal control u∗ (t) for the plant which drives the system x˙1 (t) = x2 (t) (9) x˙2 (t) = −2x1 (t) + 3u(t) (10) with x1 (0) = 0, x2 (0) = 1, x1 (π/2) and x2 (π/2) are free which minimizes the the performance index given by Z 1 2 1 π/2 2 J = x1 (π/2) + u (t)dt (11) 2 2 0 1